Packed Bed Reactor. Introduction. Model Definition

Transcription

1 Packed Bed Reactor Introduction The packed bed reactor is one of the most common reactors in the chemical industry, for use in heterogeneous catalytic processes. In essence, the reactor consists of a container filled with catalyst particles. These particles can be contained within a supporting structure, like tubes or channels, or they can be packed in one single compartment in the reactor. The structure that is formatted by the packed catalyst particles makes the modeling of mass and energy transport in the reactor a challenging task. The difficulty lies in the description of the porous structure, which gives transport of different orders of magnitudes within the particles and between the particles. In most cases, the structure in between particles is described as macroporous and the particle radius can be of the order of magnitude of 1 mm. When a pressure difference is applied across the bed, convection arise in the macropores. The pores inside the catalyst particles form the microstructure of the bed. The pore radius in the particles is often between one and ten micrometer. This model presents a simple and fast alternative for studying microscale and macroscale mass balances in packed beds and other heterogeneous reactors with bimodal pore distribution. Using a multigeometry approach, the model provides the mass and reaction distributions along the reactor and within each catalyst pellet along the reactor length. This makes it possible to evaluate the utilization of catalyst load, optimal pellet size, or inlet temperature. In this example, volatile organic compounds (VOC) and CO are oxidized in a catalytic converter. Propylene is used as representative for hydrocarbons present in the feed stream, which could be, for example, exhaust gas from a combustion process. Model Definition The following chemical reactions describe the catalytic conversion carbon monoxide and by-product hydrocarbon: CO O 2 CO 2 (8-143) PACKED BED REACTOR 613

2 9 C 3 H O 2 2 3H 2 O + 3CO 2 (8-144) For these heterogeneous catalytic reactions the rates (mol/(m 3 s)) are given by r 1 = k 1 c CO c O2 ( 1 + K CO c CO + K C3H6 c C3H6 ) 2 (8-145) r 2 = k 2 c C3H6 c O2 ( 1 + K CO c CO + K C3H6 c C3H6 ) 2 (8-146) The rate and adsorption constants are given by Arrhenius expressions: k = E A exp R g T (8-147) The values of the frequency factors and activation energies (J/mol) are taken from the literature (Ref 2) and presented in Table 8-3. TABLE 8-3: ARRHENIUS PARAMETERS RATE/ADSORPTION CONSTANT FREQUENCY FACTOR (JOULE/MOL) k k K CO K C3H ACTIVATION ENERGY (JOULE/MOL) The chemical reactions given in Equation and Equation describe a process that is limited by the second-order rate expression at high temperatures when the surface coverage is low. At low temperatures the adsorbtion reactions will be slower and limit the reaction rate due to a shortage of free catalytic sites. The pressure drop in the reactor is described by the Ergun equation dp dx ρu ( 1 Φ) = ( 1 Φ)µ ρg c D p ρu D p Φ 3 (8-148) where P is the pressure (kpa), Φ the porosity, D p the particle diameter (m), µ denotes the gas viscosity (kg/(m s)), ρ the gas density (kg/m 3 ), and x the reactor length (m). u is the reactor flow velocity (m/s) that depends on the pressure drop according to 614 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

3 C P u u feed feed = P C feed (8-149) where u feed is the inlet velocity and C the total concentration (mol/m 3 ). The mass transport in the reactor is given by the convection and diffusion equation ( D c) = R u c (8-150) where D is the diffusion coefficient (m 2 /s) and R is a source term (mol/(m 3 s)). Equation needs to be solved for all species participating in Equation and Equation Each equation also needs proper boundary conditions. At the inlet the reactant concentration is known: c CO = c COin (8-151) c C3H6 = c C3H6in (8-152) c O2 = c O2in (8-153) c CO2 = 0 c H2O = 0 (8-154) (8-155) At the outlet, the convective flux condition will let the product and reactants leave with the flow. n ( D c) = 0 (8-156) At first glance, these equations appear straight forward, especially if the velocity vector is given by an analytical expression, which is the case for plug flow in a packed bed reactor. However, the source term, R, depends on the transport inside the catalyst particles. The molar flux at the outer surface of the particles multiplied by the available outer surface area of the particles per unit volume gives the proper source term: R = A p ( N n) at r = r p (8-157) In this equation, N denotes the flux inside the porous particle (mol/(m 2 s)) and n is the outward unit vector normal to the particle surface. Equation is only valid at the particle surface, where the independent radius variable r (introduced below) equals the particle radius, r p. Furthermore, A p, denotes the pellet surface to volume ratio (m 2 /m 3 ). This property is related to the pellet radius as PACKED BED REACTOR 615

4 A p = r ---- p 3 (8-158) To properly calculate R, a mass balance is required for the catalyst pellet interior. Such a mass balance can be expressed as ( D pc c p ) + R p = 0 for x 2 y < r p (8-159) Here, D pc is the effective diffusion coefficient in the particle, c p is the concentration in the particle, and R p is the reaction rate for the heterogeneous reaction in the particle. In the catalyst pores, transport takes palace by diffusion only. Equation gives mass balance for each species for the catalyst pellet. Because the pellets are spherical, it is convenient to express these mass balances in spherical coordinates. Equation then becomes ( r 2 D pc c p ) + r 2 R p = 0 for 0 < r < r p (8-160) where r (m) is the independent variable for the position along the radius of the particle. The equation is scaled with the particle radius to get a geometry with a lower aspect ratio. This gives the final formulation: ---- D pc c p + r 2 R p = 0 for 0 < r < 1 r 2 r p 2 (8-161) The diffusion-reaction equation, combined with the boundary conditions for the particle, give the concentration distribution in the particle. The boundary conditions are: D pc c p n = 0 at r = 0 c p = εc at r = 1 (8-162) (8-163) where ε denotes the porosity of the particle, and c and c p represent the species concentrations. This implies symmetry at the center of the particle. In addition, the concentration at the surface of the particle is equal to the concentration outside the particle compensated to account for the part of the particle volume that is occupied by solid catalyst support. 616 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

5 The concentration distribution in the particle gives the molar flux at every point in the particles. This implies that the source term for the catalyst bed is given by the solution of the microscale mass balance: R = A p ( D pc c p n) (8-164) A complication in solving the derived system of equations is that the macroscale and microscale balances are defined in different coordinate systems. This problem is general for many chemical reaction engineering applications and is often solved by using an analytical approximation of the solution to the microscale balance. However, such an approach cannot be used for complicated reaction mechanisms involving several reaction species. The solution exemplified here is general and can be used for very complex reaction mechanisms involving a large number of species. Results and Discussion Figure 8-72 shows the concentration of reacting species in the reactor as function of position. The levels of carbon monoxide and propylene are significantly reduced. Figure 8-72: The concentration of reactants and products along the reactor length. The pellet radius r p is 2.5mm. PACKED BED REACTOR 617

6 The concentration can also be evaluated within each pellet, revealing whether or not the catalyst comes to efficient use. Figure 8-73 shows an example of this. r x Figure 8-73: The concentration of in-pellet CO as function of reactor position. The scaled pellet radius is given on the r-axis and the reactor position along the x-axis. From Figure 8-73 it is clear that the CO concentration is low at the center of the pellet at all reactor positions. This means that reactor performance is limited by diffusion within the pellets and that active catalyst material in the center of the pellet will never 618 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

7 come to use. Reducing the pellet diameter could potentially remedy this situation, and simulation results shown in Figure 8-74 confirm this notion. Figure 8-74: The concentration of in-pellet C 3 H 6 at three different reactor positions (5, 15, and 25 cm). Solid lines represents a pellet radius of 2.5 mm and lines with triangular markers represents a pellet radius of 1.8 mm. With a pellet radius of 2.5 mm, the reactor is limited by the in-pellet diffusion, as is indicated by the slope of the solid lines. The effect of this is that the active sites in the interior of the pellets are not used to their full potential. If the pellet size is reduced (represented by the lines with a triangular marker), the lines level out faster, suggesting larger reaction limited regions. The effect on C 3 H 6 conversion in the simulated case is an improvement from 94% to 97%. References 1. S.H. Oh, J.C. Cavendish, and L.L. Hegedus. Mathematical modeling of catalytic converter lightoff: Single-pellet studies, AIChE Journal, vol. 26 (6), pp J.B. Rawlings and J.G. Ekerdt, Chemical Reactor Analysis and Design Fundamentals, Nob Hill Publishing, Madison, PACKED BED REACTOR 619

8 Model Library path: Chemical_Engineering_Module/ Transport_and_Reactions/packed_bed_reactor Modeling Using the Graphical User Interface The approach in this example is to treat the bed as a 1D model with independent variable x. At each x-coordinate, there is a model of a single pellet representative of all pellets at that position in the packed bed. Because the pellets are spherical and the surface concentrations can be considered constant around a particle, a 1D model with independent variable r is used for each particle. To model pellets along the packed bed, many such 1D models can be placed side by side, effectively creating a 2D geometry with independent variables x and r. In the resulting model, all pellets can be modeled independently. There is therefore a 2D geometry for the pellets and a 1D geometry for the packed bed. The model involves a two-way coupling: the packed bed concentration enters the boundary condition for the particles, and the source term in the packed bed involves the flux at the particle surfaces. MODEL NAVIGATOR 1 Start COMSOL Multiphysics. 2 In the Model Navigator, click the Multiphysics button and select 2D in the Space dimension list. 3 Click the Add Geometry button and enter Particles in the Geometry name edit field. 4 Change the independent variables to x r z, then click OK to close the dialog box. 5 Select the application mode COMSOL Multiphysics>Chemical Engineering Module>Mass Transport>Diffusion. Enter C3H6p COp CO2p H2Op O2p in the Dependent variables edit field. Click the Add button. 6 Click the Add Geometry button and type Reactor in the Geometry name edit field. Also set the Space dimension to 1D. Close the dialog box by clicking OK. 7 Highlight the application mode COMSOL Multiphysics>Chemical Engineering Module>Mass Transport>Convection and Diffusion. Enter C3H6 CO CO2 H2O O2 in the Dependent variables edit field. 8 Click the Add button. 620 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

9 9 Select the application mode COMSOL Multiphysics>PDE Modes>PDE, Coefficient Form. In the Dependent variables edit field, type P. 10 Click OK to close the Model Navigator. GEOMETRY MODELING 1 From the Multiphysics menu, choose Reactor: Convection and Diffusion (chcd). 2 Hold the Shift key and click the Line button on the Draw toolbar to open the Line dialog box. 3 In the x edit field, type 0 0.3, then click OK to close the dialog box. 4 Click the Zoom Extents button on the Main toolbar. OPTIONS AND SETTINGS 1 From the Options menu, choose Constants. 2 Click the Import Variables From File button in the lower left of the Constants dialog box. PACKED BED REACTOR 621

10 3 Select the file packed_bed_reactor_const.txt (located in the same Model Library folder as the MPH-file). Alternatively, enter the following parameter names and values in the Constants dialog box: NAME EXPRESSION DESCRIPTION P_feed 2[atm] Feed Pressure T 508[K] Reactor Temperature R 8.314[Pa*m^3/mol/K] Gas constant kmco 3.90[cm/s] External mass transfer coefficient for CO kmo2 4.07[cm/s] External mass transfer coefficient for O2 kmc3h6 3.90[cm/s] External mass transfer coefficient for CO D_CO [cm^2/s] Diffusion coefficient for CO D_O [cm^2/s] Diffusion coefficient for O2 D_C3H [cm^2/s] Diffusion coefficient for C3H6 D_prod [cm^2/s] E [K] Activation energy for R1 E [K] Activation energy for R2 A1 7.07e19[cm^3/mol/s] Pre exponential factor for reaction 1 A2 1.47e21[cm^3/mol/s] Pre exponential factor for reaction 2 Aco 8.099e6[cm^3/mol] Eq constant for adsorption term Ac3h e8[cm^3/mol] Eq constant for adsorption term Eco -409[K] Activation energy for adsoption term Ec3h6 191[K] Activation energy for adsoption term u_feed 75[cm/s] Linear inflow velocity dhr e3[cal/mol] Heat of Reaction for R1 dhr e3[cal/mol] Heat of Reaction for R2 Cp 0.25[cal/g/K] Effective Heat capacity for bed kp 0.779[W/(m*K)] Thermal conductivity (ripped from other model) por_b 1-rho_b/rho_p Porosity of the bed (void/total) rho_b 0.51[g/cm^3] Density of bed 622 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

11 NAME EXPRESSION DESCRIPTION rho_p 0.68[g/cm^3] Density of pellet Rr 5[cm] reactor radius Ra pi*rr^2 Reactor area eta 0.028e-2[g/cm/s] Fluid viscosity rho_feed 1.292[kg/m^3] Inlet gas density Ctot_feed P_feed/R/T Total feed concentration x_co_feed Molar fraction of CO in feed x_o2_feed Molar fraction of O2 in feed x_c3h6_feed Molar fraction of C3H6 in feed Cinert_feed (1-x_CO_feed-x_O2_f Concentration of inert in feed eed-x_c3h6_feed)*ct ot_feed Ap 3/rp Surface area of particles per unit volume rp 0.25[cm] particle radius 4 Choose Options>Expressions>Global Expressions. 5 Click the Import Variables From File button in the lower left of the Global Expressions dialog box and select the file packed_bed_reactor_expr.txt (located in the same Model Library folder as the MPH-file). Alternatively, enter the following parameter names and values in the Global Expressions dialog box: NAME EXPRESSION DESCRIPTION Ctot (Cinert_feed+CO+O2+ Total concentration CO2+C3H6+H2O) k1 A1*exp(-E1/T) Rate constant for reaction 1 k2 A2*exp(-E2/T) Rate constant for reaction 2 KCO Aco*exp(-Eco/T) Adsorbtion constant for CO KC3H6 Ac3h6*exp(-Ec3h6/T) Adsorbtion constant for C3H6 R1 k1*cop*o2p/ Rate expression for reaction 1 (1+KCO*COp+KC3H6*C3 H6p)^2 R2 k2*c3h6p*o2p/ Rate expression for reaction 2 (1+KCO*COp+KC3H6*C3 H6p)^2 u u_feed*(ctot/ Ctot_feed)*(P_feed/ P) Linear velocity PACKED BED REACTOR 623

12 NAME EXPRESSION DESCRIPTION beta G*(1-por_b)/ Ergun equation constant (rho_feed*2*rp*por_ b^3)*(150*(1-por_b) *eta/(2*rp)+1.75*g) G u_feed*rho_feed superficial mass velocity F0 Ctot_feed*u_feed*Ra Inlet molar flow F Ctot*u*Ra Molar flow [mol/s] 6 Click OK. PHYSICS SETTINGS Subdomain Settings Convection and Diffusion 1 From the Multiphysics menu, select 2 Reactor: Convection and Diffusion (chcd). 2 From the Physics menu, select Subdomain Settings. 3 Select Subdomain 1 in the Subdomain Settings dialog box. 4 Specify the settings as in the following table. SETTINGS C3H6 CO CO2 H2O O2 D D_C3H6 D_CO D_prod D_prod D_O2 R 5 Click the Init tab and type 1e-6 in all edit fields. 6 Click OK. -Ap*C3H6fl ux Boundary Conditions Convection and Diffusion 1 From the Physics menu, choose Boundary Settings. 2 Select Boundary 1. -Ap*COflux -Ap*CO2flu x -Ap*H2Oflu x u u u u u u -Ap*O2f lux 3 Enter the species concentrations of the inlet stream. For each species in the table, find the Boundary condition list and select Concentration. Type in the entries from the following table: BOUNDARY C3H6 CO CO2 H2O O2 1 x_c3h6_feed* Ctot_feed 4 Select Boundary 2. x_c3h6_feed *Ctot_feed 0 0 x_c3h6_feed *Ctot_feed 624 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

13 5 Set the Boundary condition to Convective flux for all species on the outlet boundary. 6 Click OK. Subdomain Settings PDE, Coefficient Form 1 From the Multiphysics menu, select 3 Reactor: Coefficient Form (c). 2 From the Physics menu, select Subdomain Settings. 3 Select Subdomain 1 in the Subdomain Settings dialog box. 4 Specify the settings as in the following table. SETTINGS f VALUE Px+beta*(F/F0)*(P_feed/P) 5 Click the Init tab and type 1e5 in all edit fields. 6 Click OK. Boundary Conditions PDE, General Form 1 From the Physics menu, choose Boundary Settings. 2 Select Boundary 1, then type 1 in the h edit field. 3 In the R edit field, type P_feed. 4 Select Boundary 2, then click the Neumann boundary condition button. 5 Click OK. The source terms in the reactor mass balances depend on fluxes from the catalyst pellets. To know this the catalyst model must also be set up. Once this is done the reaction rate can be calculated by coupling the two models. GEOMETRY MODELING 1 From the Multiphysics menu, choose 1 Particles: Diffusion (chdi). 2 Draw a unit square with its lower left corner at (0,0). 3 Click the Zoom Extents button on the Main toolbar. PHYSICS SETTINGS Subdomain Settings Convection and Diffusion 1 From the Physics menu, select Subdomain Settings. 2 Select Subdomain 1 in the Subdomain Settings dialog box. PACKED BED REACTOR 625

14 3 Specify the settings according to the following table. SETTINGS C3H6P COP CO2P H2OP O2P D (anisotropic) (D_C3H6/ rp^2)*r^2 4 On the Init page, type 1e-6 in all edit fields. 5 Click OK. Boundary Conditions Diffusion 1 From the Physics menu, choose Boundary Settings. 2 Select Boundary (D_CO/ rp^2)*r^ (D_prod/ rp^2)*r^2 R -r^2*r2 -r^2*r1 r^2*(r1+3* R2) (D_prod/ rp^2)*r^2 r^2*3*r (D_O2/ rp^2)*r ^2 -r^2*(0.5*r1+4.5*r2) 3 Enter the species concentration on the pellet surface. This can be taken from the reactor model through a coupling variable. For now, find the Boundary condition list and select Concentration, then type the entries from the following table: BOUNDARY C3H6P COP CO2P H2OP O2P 3 C3H6bulk CObulk CO2bulk H2Obulk O2bulk 4 Leave Boundaries 1, 2, and 4 with the default boundary condition Insulation/ Symmetry. 5 Click OK. The species concentration at the pellet boundary can be taken from the reactor model. Each position along the x-axis, in the reactor model, corresponds to a position along the x-axis in the pellet model. So the concentration at pellet-bulk interface can be taken form the reaction model with a Subdomain Extrusion coupling variable. This concentration at the pellet surface will give rise to mass flux at the interface. This mass flux can be coupled back to give the total reactor reaction rate for all species. To set up the coupling follow these steps: 1 From the Multiphysics menu, select 2 Reactor: Convection and Diffusion (chcd). 2 Select Options>Extrusion Coupling Variables>Subdomain Variables. 626 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

15 3 Define the coupling variables by entering names and expressions according to the following table: NAME CObulk O2bulk CO2bulk C3H6bulk H2Obulk EXPRESSION CO O2 CO2 C3H6 H2O 4 For each entry also select General transformation. This must be done for all 5 species. 5 Click the Destination tab. 6 In the Geometry list, choose Particles. Make sure that the Level list is set to Boundary. 7 Select CObulk in the Variable list. 8 Select Boundary 3 9 Enter x*0.30 in the Destination transformation edit field. This correspond to that the reactor length is only 30% of the pellet domain because the pellet domain is scaled. 10 Repeat Steps 8 and 9 for all other species in the Variable list. 11 Click OK. To uphold the concentration constraint on the particle surface there will be a diffusive flux out through the top boundary of the pellet. This flux is equal to the total reaction rate if it is multiplied by the specific surface area, at steady state. But to make this flux available this must be coupled back to the reactor geometry. 1 From the Multiphysics menu, select 1 Particles: Diffusion (chdi). 2 Select Options>Extrusion Coupling Variables>Boundary Variables. 3 Mark Boundary 3 in the Boundary Selection list. 4 Define the coupling variables by entering names and expressions according to the following table: NAME COflux O2flux CO2flux EXPRESSION D_CO*COpr*r/rp D_O2*O2pr*r/rp D_prod*CO2pr*r/rp PACKED BED REACTOR 627

16 NAME C3H6flux H2Oflux EXPRESSION D_C3H6*C3H6pr*r/rp D_prod*H2Opr*r/rp 5 For each entry also click the General transformation button. This must be done for all 5 species. 6 Click the Destination tab. 7 In the Geometry list, choose Reactor and make sure that the Level list is set to Subdomain. 8 Select COflux in the Variable list. 9 Select Subdomain 1. In the Destination transformation edit field, type x/0.30. This transformation accounts for the fact that the reactor length is only 30% of the pellet domain because the pellet domain is scaled. 10 Repeat Steps 8 and 9 for all other species in the Variable list. 11 Click OK. MESH GENERATION 1 From the Multiphysics menu, select 2 Reactor: Convection and Diffusion (chcd). 2 From the Mesh menu, choose Free Mesh Parameters. 3 In the Maximum Element Size edit field, enter Click the Remesh button and close the dialog box. 5 From the Multiphysics menu, select 1 Particles: Diffusion (chdi). 6 From the Mesh menu, choose Mapped Mesh Parameters. 7 Click the Boundary tab. Select Boundaries 1 and 4 by Ctrl-clicking their entries in the Boundary selection list. 8 Select the Constrain element distribution check box, then enter 30 in the Number of edge elements edit field. Also enter 0.2 in the Element ratio edit field. This makes the mesh finer near the pellet surface. 9 Select Boundary 3 in the Boundary selection list. Select the Constrain element distribution check box, then enter 120 in the Number of edge elements edit field. 10 Click Remesh. When the mesher has finished, click OK to close the dialog box. Now there are equally many numbers of elements on the coupled lines, which is good for stability reasons when coupling in both directions. 628 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

17 OPTIONS AND SETTINGS To evaluate the conversion of C 3 H 6 and CO it is convenient to set up dedicated expressions for this. 1 From the Multiphysics menu, select 2 Reactor: Convection and Diffusion (chcd). 2 From the Options menu, choose Integration Coupling Variables>Boundary Variables. 3 Select 2 in the Boundary Selection list. 4 Create the following variables: NAME X_C3H6 X_CO EXPRESSION 1-C3H6/(x_C3H6_feed*Ctot_feed) 1-CO/(x_CO_feed*Ctot_feed) 5 Click OK. COMPUTING THE SOLUTION To save memory the Stationary Segregated Solver will be used. This solver alternates between solving for the species in the Reactor geometry and the pellet each iteration. To avoid feedback from the variables that are currently not being solved for a strict one way coupling will be enforced for the couplings between the geometries. This is not default when the coupling takes place on a boundary condition, which is the case for the Particles geometry. 1 From the Multiphysics menu, select 1 Particles: Diffusion (chdi). 2 Go to Properties in the Physics menu. 3 Set the Constraint type drop down list to Non-Ideal. Click OK to close the dialog box. 4 Click the Solver Parameters button on the toolbar. 5 Select the Stationary Segregated solver in the Solver list. Click OK to close. 6 Click the Solve button. The remaining solving stages continue after the postprocessing section. POSTPROCESSING AND VISUALIZATION To create Figure 8-72 perform the following steps 1 From the Postprocessing menu, open the Domain Plot Parameters dialog box. 2 On the General page, select the Keep current plot check box. 3 Click the Title/Axis button. 4 In the Title edit field, enter Concentration along reactor length. PACKED BED REACTOR 629

18 5 Enter Reactor length, [m] in the First axis label and Concentration, [mol/ m³] in the Second axis label edit field. 6 Click the Line/Extrusion tab. 7 Select Concentration, C3H6 in the Predefined quantities list. 8 Click the Line Settings button. Select Color in the Line color list. 9 Click the Color button and select black. Click OK twice to close the dialog boxes. 10 Click Apply to plot the concentration. 11 Repeat Steps 7 to 10 for the species CO, CO 2, and O 2 giving each line a different color. 12 Click OK. To create Figure 8-73 preform the following steps 1 From the Postprocessing menu open the Plot Parameters dialog box. 2 Click the Surface tab. 3 Select Concentration, O2p in the Predefined quantities list. 4 Click the Height Data tab. 5 Select the Height data check box. 6 From the Predefined quantities list, select Concentration, O2p. Click OK. To investigate if better pellet utilization can be achieved, the pellet size is reduced. This affects the packing of the catalyst and reduces the diffusive length within the pellets. COMPUTING THE SOLUTION 1 Click the Solver Parameters button on the Main toolbar. 2 From the Solver list, select Parametric segregated. 3 To vary the pellet radius, type rp in the Parameter names edit field. 4 In the Parameter values edit field, type On the Parametric page, select the Manual tuning of parameter step size check box. 630 CHAPTER 8: TRANSPORT, REACTIONS, AND REACTION ENGINEERING

19 6 Enter the values in the following table: SETTINGS C3H6 Initial Step size Minimum Step size 1e-6 Maximum step size This ensures that the solver does not jump from to in a single step; taking small steps can increase the solver stability. 7 Click OK to close the Solver Parameters dialog box. 8 Click the Restart button on the Main toolbar. POSTPROCESSING AND VISUALIZATION To reproduce the plot in Figure 8-74, perform the following steps: 1 From the Postprocessing menu, open the Cross-Section Plot Parameters dialog box. 2 On the General page, select in the Solutions to use list. 3 Select the Keep current plot check box. 4 On the Line/Extrusion page, select Concentration, C3H6 from the Predefined quantities list. 5 In the Cross-section line data area, set x1 to 0 and r1 to 1. 6 Select the Multiple parallel lines check box. 7 Set the Number of lines to 3, then click Apply. 8 On the General page, select in the Solutions to use list. 9 Click OK. PACKED BED REACTOR 631